zbMATH — the first resource for mathematics

Characterizing certain incomplete infinite-dimensional absolute retracts. (English) Zbl 0629.54011
Let $$\sigma =\{t\in [-1,1]^{\infty}| t_ i=0$$ for all but finitely many $$i\}$$ and let $$\Sigma =\{t\in Q^{\infty}| t_ i=0$$ for all but finitely many $$i\}$$ where $$Q=[-1,1]^{\infty}$$. The second author has obtained characterization theorems for $$\sigma$$ and $$\Sigma$$ [Proc. Am. Math. Soc. 92, 111-118 (1984; Zbl 0577.57005)]. The authors obtain characterizations for $$\sigma$$ and $$\Sigma$$ as corollaries of a more general characterization theorem for certain incomplete subsets of Hilbert spaces. This more general characterization theorem also yields a characterization of $$\Sigma\times \sigma$$ and gives characterizations of certain absolute Borel sets and of certain ARs. A triangulation theorem for the spaces characterized is also obtained. One version of a general characterization theorem obtained is as follows. If $${\mathcal C}$$ is an additive topological class hereditary with respect to closed subsets, and if there exists a $${\mathcal C}$$ absorbing set $$\Omega$$ in a Hilbert space, then an AR X is homeomorphic to $$\Omega$$ if and only if X is $${\mathcal C}_{\sigma}$$, X is strongly $${\mathcal C}$$ universal and $$X=\cup^{\infty}_{i=1}X_ i$$ where each $$X_ i$$ is a strong Z-set in X. X is strongly $${\mathcal C}$$ universal if for each map f from C to X, where $$C\in {\mathcal C}$$, for each closed subset D of C such that $$f|_ D$$ is a Z-embedding, and for each open cover $${\mathcal U}$$ of X, there exists a Z-embedding h from C into X such that $$h|_ D=f|_ D$$ and such that f and h are $${\mathcal U}$$ close.
Reviewer: D.J.Garity

MSC:
 54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) 54F65 Topological characterizations of particular spaces 54C50 Topology of special sets defined by functions
Full Text: